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March 3rd, 2014, 12:41 PM   #1
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limit involving 2 sequences

Hello!
I have encountered the following problem involving some limits:
Let The problems asks this limit:

Well I have tried several ways like Cesaro Stolz but it didn't work. After that I've tried to use a matrix:Let A be
Then But I couldn't solve the problem...
Any help would be appreciated!
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March 3rd, 2014, 05:33 PM   #2
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Re: limit involving 2 sequences



If you can do an series expansion at n = ? of you've got most of a proof.
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March 3rd, 2014, 10:08 PM   #3
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Re: limit involving 2 sequences

Quote:
Originally Posted by greg1313


If you can do an series expansion at n = ? of you've got most of a proof.
Unfortunately I haven't reached this level yet. I don't know how to do a series expansion and I was thinking there might be some other way to solve this exercise.
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March 5th, 2014, 10:41 AM   #4
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Re: limit involving 2 sequences

How can I actually proof that using what you told me? Do you mean that the limit is actually e?
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March 5th, 2014, 12:11 PM   #5
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Re: limit involving 2 sequences

Writing out the first few terms of the series, it becomes apparent that a_n/b_n = 1, 1, 1/2, 1/3, 1/4, 1/5, 1/6 . . .

We may thus write the limit as



Using Stirling's approximation for n!, we have

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March 5th, 2014, 08:33 PM   #6
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Re: limit involving 2 sequences

Unfortunately, only testing some terms of doesn't establish proof that is is true for all values of n. Is it possible to show this limit analytically? My first few attempts tonight were unsuccessful.
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March 6th, 2014, 04:12 AM   #7
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Re: limit involving 2 sequences

I'm sorry, I should have mentioned that is not a complete proof. My apologies if I haven't been of much assistance.
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March 6th, 2014, 07:12 AM   #8
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Re: limit involving 2 sequences

I claim that



Proof:




Now it needs to be shown (using the induction hypothesis) that


Now we may write the limit as



Using Stirling's approximation for n!, we have



That's about as analytic as I can get at this time and I can see why there might be problems. My logic is that, for n sufficiently large, the limit may be written as immediately above. Then,



So (with L'Hopital's rule),

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March 6th, 2014, 09:10 AM   #9
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Re: limit involving 2 sequences

Thank you! After I wrote that I calculated that limit without using Stirling's approximation. I used d'Alembert's principle and after some arrangements I got :So this is e. Thanks again!
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